Consider a spherically symmetric system of clouds, extending from
rin to rout.
The mass of the individual clouds is conserved, but it is not
necessarily the same
for all clouds. The following physical parameters are all functions of
r: the cloud
number density nc(r), the cloud particle density
N, the cloud emissivity jc(r),
and the cloud velocity v(r).

Most cases of interest involve optically thick clouds, where energy
conservation
requires that the total emergent line and diffuse continuum luminosity
equal the energy absorbed by the gas. In this case, the geometrical cross
section of the cloud as seen from the center,
Ac(r), is another useful parameter.

Designate
l(r)
as the flux emitted by the cloud, in a certain emission line l,
per unit projected surface area (erg s-1
cm-2), we have the following relation
for the cloud emission:

(49)

We now make the following simplified assumptions about the radial
dependence of the different parameters:

(50)

(51)

(52)

(53)

and

(54)

Using these definitions, the radial dependence of the ionization
parameter is:

(55)

5.1.1. Covering factor. For optically thick
clouds, it is useful to introduce the
covering factor, C(r), which is the fraction of sky covered by
photoionized gas clouds, as seen from the center. For a single cloud

(56)

and for a thin shell of thickness dr

(57)

The integrated covering factor can be estimated from the Lyman continuum
observations (chapter 2) or from comparing the line and continuum
luminosities
(chapter 6). For the more luminous AGNs it is
about 0.1 for the BLR and less
than that for the NLR. It is thus justified to neglect obscuration and
proceed
on the assumption that the flux reaching the clouds depends only on
r.

There are interesting cases where the number of radius dependent parameters
is considerably reduced. An important example is a system of spherical
clouds, of radius Rc(r), in pressure equilibrium with
a confining medium of
pressure P. The kinetic temperature of a photoionized gas is only
a weak function of U, and we can safely assume that the radial
dependences of P and N
are identical (r-s). Since R3cN =
const., the cloud cross-section is

(59)

(i.e. s = -3/2q) and the column density is

(60)

Assume further that the clouds are moving in or out with their
virial velocity.
Mass conservation (ncvr2 =
const.) requires that p = 2-t and substituting
t = 1/2 we get the following radial dependence of the covering
factor:

(61)

General considerations (chapter 9) suggest
that 1 s 5/2 in many cases of
interest. In particular, s = 9/4 corresponds to a case where the
covering factor of a thin shell is proportional to the shell thickness.

The total flux, in all forms of emission, from a shell of
thickness dr is proportional to dC(r)(4), and the cumulative flux is
proportional to

(62)

This is, in fact, a good approximation for many individual lines
(i.e. m = 2 is a good approximation for many lines).

We conclude that in those cases with s > 3/4, the cumulative
covering
factor, and therefore the integrated line emission, increases rapidly
outward,
and the innermost parts of the atmosphere do not contribute much to the
line emission. For s
3/4 much of the line emission takes place close to the
center and there is a natural boundary to the cloud distribution. Note,
however,
some potential difficulties. For example, the column density of ionized
material
depends on U and r, and clouds that are optically thick
near rout may become transparent closer in, or vice versa.

5.1.3. Specific models.Fig. 11 shows cumulative BLR line fluxes, as a
function
of rout for a constant ionization parameter model,
s = 2. They were calculated
as explained in chapter 4, and added
according to the prescription explained
above. In this particular case, L(ionizing continuum) =
1046erg s-1U = 0.3
for all clouds and the normalization is such that Ncol
= 2 × 1023cm-2 where
N = 1010cm-3. The integration
starts at rin = 1016.5cm, where the
density is 1012cm-3, and carried out to
large radii. Vertical cuts in the diagram
correspond to a certain rout and give the integrated
line fluxes up to that radius.

Figure 11. Cumulative broad line fluxes, as
a function of rout, for a model with U =
0.3 and s = 2. The model is calculated for the continuum shown
in Fig. 7, assuming
L(ionizing luminosity) = 1046erg
s-1. the value of rout for other
luminosities is obtained
by noting that routL1/2.

The narrow line fluxes can be integrated in a similar way, and
Fig. 12 shows
a model with s = 1, normalized such that U = 3.6 ×
10-2 where N = 105cm-3
and Ncol = 1022 cm-2. Here again L(ionizing luminosity) = 1046erg s-1, the
inner radius is at 1019.5cm and the density there is
106cm-3. Note that here,
and in the BLR model of Fig. 11, the line
intensity shown is normalized in such
a way that the covering factor is unity at the outermost radius shown.

Figure 12. Cumulative narrow line fluxes,
as a function of rout, for a model with s = 1
and parameters as explained in the text.